#1043441

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2678	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3318 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72329291, #72329339, #72329402
maximal normal subgroup of	#72329291, #72329339, #72329402
maximal quotient of	#72330758

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 950637);