#7670016

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7124 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,d^c=dh,e^c=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281938, #72282224
maximal normal subgroup of	#72281938, #72282224

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7577212);