#7682723

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

8561 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312909, #72342314, #72342320
maximal normal subgroup of	#72312909, #72342314, #72342320

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7589919);