#7604338

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518936
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#28231	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1894 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=f,c^2=hi,d^2=gi,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280838, #72284652, #72285302
maximal normal subgroup of	#72280838, #72284652, #72285302

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7511534);