#6373142

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×#120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2274 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

(1,2)(3,4)(5,6)(7,12)(8,10)(9,11)(13,14,18,22,17,21,25,28)(15,23,24,16,20,19,27,26);(5,7,9,8)(6,10,11,12)(13,15,17,20)(14,19,21,23)(16,22,26,28)(18,24,25,27);(2,4);(5,8,9,7)(6,10,11,12)(13,16,25,23,17,26,18,19)(14,20,28,24,21,15,22,27);(15,20)(16,26)(19,23)(24,27)

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=fh,h^2=i,b^a=bf,c^a=cg,e^b=ei,e^d=ei

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307281, #72307303, #72307307, …
maximal normal subgroup of	#72307281, #72307303, #72307307, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280338);