#346546

order	512
nilpotency class	3
isoclinism	isoclinic class of #346518
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62988	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

3229 conjugacy classes of subgroups; only the 224 normal subgroups are shown.

normal subgroups only

This group has 3229 conjugacy classes of subgroups — too many to draw. Only its 224 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2,5,8)(3,6,4,7)(9,10,14,13)(11,15,19,18)(12,16,21,20)(17,23,24,22);(2,6)(3,4)(7,8)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24);(9,12)(10,16)(11,17)(13,20)(14,21)(15,22)(18,23)(19,24);(2,7)(3,4)(6,8)(9,13,14,10)(11,18,19,15)(12,20,21,16)(17,23,24,22)

Polycyclic presentation:

2^9|a^2=hi,d^2=i,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,e^d=eh,g^d=gh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72341912, #72341967, #72341997
maximal subgroup of	#72231727, #72268303, #72268305, …
maximal normal subgroup of	#72231727, #72268303, #72268305, …
maximal quotient of	#72231593, #72231596, #72231599, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253742);