#347057

order	512
nilpotency class	3
isoclinism	isoclinic class of #347042
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	C₂⁵	C₂⁴
prime radical	#62824	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2563 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 2563 conjugacy classes of subgroups — too many to draw. Only its 192 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) = 32

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249447, #72258573, #72266832, …
maximal normal subgroup of	#72249447, #72258573, #72266832, …
maximal quotient of	#72258580

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254253);