#7732835

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

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

7230 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238148, #72238200, #72238223
maximal normal subgroup of	#72238148, #72238200, #72238223

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7640031);