#7837521

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

7533 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)(13,14,18,15)(16,21,24,19)(17,20,25,22)(23,27,28,26);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,15)(16,19,24,21)(17,22,25,20)(23,27,28,26);(6,8)(10,12)(14,15)(17,25)(19,21)(23,28);(7,11)(10,12)(13,15,18,14)(16,19,24,21)(17,22,25,20)(23,26,28,27);(5,7)(6,10)(8,12)(9,11)(13,16)(14,19)(15,21)(17,23)(18,24)(20,26)(22,27)(25,28);(5,8,9,6)(7,12,11,10)(13,17,18,25)(14,20,15,22)(16,23,24,28)(19,26,21,27)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238194, #72238660, #72257158, …
maximal normal subgroup of	#72238194, #72238660, #72257158, …
maximal quotient of	#72256820

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7744717);