#7185561

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#30529	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1864 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7092757);